Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

11.19 Evaluate the surface integral

∫

r·dS,whereris the position vector, over that part

of the surfacez=a^2 −x^2 −y^2 for whichz≥0, by each of the following methods.

(a) Parameterise the surface asx=asinθcosφ,y=asinθsinφ,z=a^2 cos^2 θ,

and show that

r·dS=a^4 (2 sin^3 θcosθ+cos^3 θsinθ)dθ dφ.

(b) Apply the divergence theorem to the volume bounded by the surface and

the planez=0.

11.20 Obtain an expression for the valueφPat a pointPof a scalar functionφthat
satisfies∇^2 φ= 0, in terms of its value and normal derivative on a surfaceSthat
encloses it, by proceeding as follows.
(a) In Green’s second theorem, takeψat any particular pointQas 1/r,wherer
is the distance ofQfromP. Show that∇^2 ψ= 0, except atr=0.
(b) Apply the result to the doubly connected region bounded bySand a small
sphere Σ of radiusδcentred on P.
(c) Apply the divergence theorem to show that the surface integral over Σ
involving 1/δvanishes, and prove that the term involving 1/δ^2 has the value
4 πφP.
(d) Conclude that

φP=−

1

4 π

∫

S

φ

∂

∂n

(

1

r

)

dS+

1

4 π

∫

S

1

r

∂φ

∂n

dS.

This important result shows that the value at a pointPof a functionφ

that satisfies∇^2 φ= 0 everywhere within a closed surfaceSthat enclosesP

may be expressedentirelyin terms of its value and normal derivative onS.

This matter is taken up more generally in connection with Green’s functions

in chapter 21 and in connection with functions of a complex variable in

section 24.10.

11.21 Use result (11.21), together with anappropriately chosen scalar functionφ,to
prove that the position vector ̄rof the centre of mass of an arbitrarily shaped
body of volumeVand uniform density can be written

̄r=

1

V

∮

S

1

2 r

(^2) dS.
11.22 A rigid body of volumeVand surfaceSrotates with angular velocityω. Show that
ω=−

1

2 V

∮

S

u×dS,

whereu(x) is the velocity of the pointxon the surfaceS.
11.23 Demonstrate the validity of the divergence theorem:

(a) by calculating the flux of the vector

F=

αr

(r^2 +a^2 )^3 /^2

through the spherical surface|r|=

√

3 a;

(b) by showing that

∇·F=

3 αa^2

(r^2 +a^2 )^5 /^2

and evaluating the volume integral of∇·Fover the interior of the sphere

|r|=

√

3 a. The substitutionr=atanθwill prove useful in carrying out the

integration.